The question is asking you to build the "quadratic formula" from scratch.
The quadratic formula is one recipe to find values of x in a quadratic equation.
Completing the square is a different recipe.
Since they both give the same answers, it should be possible to go from one to the other.
A square has the form:
(x+k)^2 = x^2 + 2kx + k^2
As you can see, it is best suited for "monic" quadratics (fancy word to mean that the first coefficient is a 1)
So, first step, divide your whole equation by "a"
(a/a)x^2 + (b/a)x + (c/a) = 0/a
a/a is the same as 1 and 0/a is the same as 0
x^2 + (b/a)x + (c/a) = 0
This is now a "monic" quadratic and it will have the same x values as the original one (because it is equivalent)
Next, we have to force it to look like the square:
x^2 + 2kx + k^2
so we need to make 2k = b/a
this forces us to use k = b/2a
which then gives us
k^2 = (b/2a)^2 = (b^2) / 4a^2
but the equation has a "c/a" instead of the value of k^2
so we move the c/a to the other side
x^2 + (b/a)x + 0 = -(c/a)
and add [b^2 / 4a^2] to both sides
x^2 + (b/a)x + b^2/4a^2 = b^2/4a^2 - c/a
The left side first:
We know the left side is a square because we just busted our derrieres to make it so.
(x + (b/2a))^2 = b^2/4a^2 - c/a
Next, the right side. First, put everything over the same denominator (4a^2)
(x + (b/2a))^2 = (b^2 - 4ac)/4a^2
Now, square root both sides, remembering tha the result of a square root can be positive or negative;
for example, the square root of +4 could be -2, because (-2)^2 = +4.
x + b/2a = +/- sqrt(b^2 - 4ac) / 2a
+/- means "plus or minus"
Move the b/2a to the right:
x = -b/2a +/- sqrt(b^2 - 4ac)/2a
Factor out the denominator (it is the same 2a for both terms)
x = [ -b +/- sqrt( b^2 - 4ac ) ] / 2a